A143107 -id:A143107 - cp's OEIS frontend

A143108 Let H(2,d) be the space of polynomials p(x,y) of two variables with nonnegative coefficients such that p(x,y)=1 whenever x + y = 1. a(n) is the number of different polynomials in H(2,d) with exactly n distinct monomials and of maximum degree minus 1, i.e., of degree 2n-4.

0, 0, 3, 4, 10, 24, 32, 56

Offset: 1

Jiri Lebl (jlebl(AT)math.uiuc.edu), Jul 25 2008

a(n) is also the number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n states and of degree 2n-4. - Carlos Améndola, Aug 05 2025

Carlos Améndola, Viet Duc Nguyen, and Janike Oldekop, One-dimensional Discrete Models of Maximum Likelihood Degree One, arXiv:2507.18686 [math.ST], 2025. See p. 24.
John P. D'Angelo, Simon Kos and Emily Riehl, A sharp bound for the degree of proper monomial mappings between balls, J. Geom. Anal., 13(4):581-593, 2003.
John P. D'Angelo and Jiří Lebl, Complexity results for CR mappings between spheres, arXiv:0708.3232 [math.CV], 2008.
John P. D'Angelo and Jiří Lebl, Complexity results for CR mappings between spheres, Internat. J. Math. 20 (2009), no. 2, 149-166.
Jiří Lebl and Daniel Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 [math.CV], 2008-2010.
Jiří Lebl and Daniel Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, Linear Algebra Appl., 433 (2010), no. 4, 824-837

Cf. A143107, A143109.

Mathematica
```
See the paper by Lebl-Lichtblau.
```

Possibly can be computed from A143107 except for the third term, but this is not proved. Let b_n be elements of A143107, then a_n = 2 ( b_2 b_{n-1} + b_3 b_{n-2} + ... + b_{n-1} b_2 ).

A143109 Let H(2,d) be the space of polynomials p(x,y) of two variables with nonnegative coefficients such that p(x,y)=1 whenever x + y = 1. a(n) is the number of different polynomials in H(2,d) with exactly n distinct monomials and of maximum degree minus two, i.e., of degree 2n-5.

Original entry on oeis.org

0, 0, 0, 11, 38, 88, 198

Offset: 1

Jiri Lebl (jlebl(AT)math.uiuc.edu), Jul 25 2008

It is unknown but conjectured that this is a sequence of finite numbers. Note that if we went one degree lower and look at polynomials of degree 2n-6, then there are infinitely many if any exist in H(2,d).

Likely an erroneous version of A387029. - Sean A. Irvine, Aug 13 2025

J. P. D'Angelo, Simon Kos and Emily Riehl, A sharp bound for the degree of proper monomial mappings between balls, J. Geom. Anal., 13(4):581-593, 2003.
J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, arXiv:0708.3232 [math.CV], 2008.
J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, Internat. J. Math. 20 (2009), no. 2, 149-166.
J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 [math.CV], 2008-2010.
J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, Linear Algebra Appl., 433 (2010), no. 4, 824-837

Cf. A143107, A143108, A387029.

Mathematica
```
(* See the paper by Lebl-Lichtblau. *)
```

A387029 Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n+1 states and of degree 2n-3.

Original entry on oeis.org

0, 0, 12, 38, 88, 198, 332

Offset: 1

Carlos Améndola, Aug 05 2025

A143109 is likely an erroneous version of this sequence.

Table 2 of Lebl and Lichtblau gives (incorrect) a(3)=11.

			For n=3 there are a(3)=12 models supported on 3+1=4 states of degree 2*3-3=3. Encoding each model parametrization as a bivariate polynomial shows why the 4th term of A143109 is also 12. Concretely, the following polynomials in x,y with 4 terms and of degree 2*4-5=3 yield the constant 1 when making the substitution y=1-x:
  1.  x + x^2*y + 2*x*y^2 + y^3,
  2.  x + x^2*y + y^2 + x*y^2,
  3.  x + x*y + x*y^2 + y^3,
  4.  x^2 + 2*x^2*y + 3*x*y^2 + y^3,
  5.  x^2 + 2*x^2*y + y^2 + 2*x*y^2,
  6.  x^2 + 2*x*y + x*y^2 + y^3,
  7.  x^2 + y + x^2*y + x*y^2,
  8.  x^3 + 2*x*y + x^2*y + y^2,
  9.  x^3 + 3*x^2*y + 3*x*y^2 + y^3,
  10. x^3 + 3*x^2*y + y^2 + 2*x*y^2,
  11. x^3 + y + 2*x^2*y + x*y^2,
  12. x^3 + y + x*y + x^2*y.

C. Améndola, V. Nguyen and J. Oldekop, One-dimensional discrete models of maximum likelihood degree one, arXiv:2507.18686 [math.ST] 2025.
A. Bik and O. Marigliano, Classifying one-dimensional discrete models with maximum likelihood degree one, Adv. Appl. Math., 170 (2025), 102928.
J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 [math.CV], 2008-2010.

Cf. A143107, A143108, A143109, A386841.

A143106 Odd degrees for which (up to swapping of variables) there exists a unique polynomial p(x,y), such that p(x,y)=1 when x+y=1, with positive coefficients and such that the number of terms is minimal (equal to (d+3)/2). There always exists a group invariant polynomial (see any of the references), but for many degrees, other such extremal polynomials exist.

Original entry on oeis.org

1, 3, 5, 9, 17, 21

Offset: 0

Jiri Lebl, Jul 25 2008

This sequence is a subsequence of A143105. It is unknown if this is the same sequence, nor if this sequence is infinite (conjectured to be such). It is not currently computationally feasible to find out if 21 belongs in this sequence or not.

			7 is not in the sequence as there are two noninvariant polynomials with minimal number of terms: x^7 + 7/2 xy + 7/2 x^5y + 7/2 xy^5 + y^7 and x^7 + 7 x^3y + 7 xy^3 + 7 x^3y^3 + y^7. This is beside the group invariant x^7 + 7 x^3y + 14 x^2y^3 + 7 xy^5 + y^7 (and one with x,y reversed).

J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, Internat. J. Math. 20 (2009), no. 2, 149-166.
J. P. D'Angelo and J. Lebl, Complexity results for CR mappings between spheres, arXiv:0708.3232 [math.CV], 2008.
J. P. D'Angelo, Simon Kos and Emily Riehl, A sharp bound for the degree of proper monomial mappings between balls, J. Geom. Anal., 13(4):581-593, 2003.
J. Lebl, Addendum to Uniqueness of certain polynomials constant on a hyperplane, preprint arXiv:1302:1441
J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, arXiv:0808.0284 [math.CV]
J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, Linear Algebra Appl., 433 (2010), no. 4, 824-837

Cf. A143105, A143107.

Mathematica
```
See the paper by Lebl-Lichtblau
```

Added term 21 that was recently computed, see the recent preprint by Lebl. Added publication data for Lebl-Lichblau paper. Corrected and edited by Jiri Lebl, May 02 2014

A386841 Triangle read by rows: T(n,k) is the number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on the n-dimensional probability simplex and of degree 2n-k (n>=1, 1<=k<=n).

Original entry on oeis.org

1, 1, 3, 2, 4, 12, 4, 10, 38, 82, 2, 24, 88, 254, 602, 4, 32, 198, 643, 2421, 6710, 8, 56, 332, 1442, 6445, 23285, 83906, 4

Offset: 1

Carlos Améndola, Aug 12 2025

The range of k is precisely chosen so that T(n,k) is positive. That is, whenever the degree is higher than 2n-1 or lower than n, there are no fundamental models.

			When n=1 then k=1 and the unique model T(1,1)=1 corresponds to the model described by a Bernoulli random variable that assigns probabilities 1-t and t to two possible states, 0<=t<=1. This line segment parametrizes the 1-dimensional probability simplex.
When n=2 we have 1<=k<=2. The T(2,1)=1 unique fundamental model with degree 3 corresponds to the parametrization t -> ((1-t)^3, 3t(1-t), t^3) and the T(2,2)=3 fundamental models of degree 2 correspond to the parametrizations ((1-t)^2, 2t(1-t), t^2) , (1-t, t(1-t), t^2) and ((1-t)^2, t(1-t), t).
Continuing in this way, the first five rows (1<=n<=5) of the fundamental models triangle are:
  1
  1 3
  2 4 12
  4 10 38 82
  2 24 88 254 602

Carlos Améndola, Viet Duc Nguyen, and Janike Oldekop, One-dimensional discrete models of maximum likelihood degree one, arXiv:2507.18686 [math.ST], 2025. See p. 20 (Figure 9).
Arthur Bik and Orlando Marigliano, Classifying one-dimensional discrete models with maximum likelihood degree one, Adv. Appl. Math., 170 (2025), 102928.

Columns 1..4 are A143107, A143108, A387029, A386840.

A386840 Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n states and of degree 2n-6.

Original entry on oeis.org

0, 0, 0, 0, 82, 254, 643, 1442

Offset: 1

Carlos Améndola, Aug 05 2025

Unlike A143107 and A143108 (and conjecturally A143109), there are infinitely many polynomials in H(2,d) of degree 2n-6. Nevertheless, this sequence consists of finite numbers.

C. Améndola, V. Nguyen and J. Oldekop, One-dimensional discrete models of maximum likelihood degree one, arXiv:2507.18686 [math.ST] 2025.
A. Bik and O. Marigliano, Classifying one-dimensional discrete models with maximum likelihood degree one, Adv. Appl. Math., 170 (2025), 102928.
J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a hyperplane, Linear Algebra Appl., 433 (2010), no. 4, 824-837

Cf. A143107, A143108, A143109, A387029, A386841.

cp's OEIS Frontend

A143108 Let H(2,d) be the space of polynomials p(x,y) of two variables with nonnegative coefficients such that p(x,y)=1 whenever x + y = 1. a(n) is the number of different polynomials in H(2,d) with exactly n distinct monomials and of maximum degree minus 1, i.e., of degree 2n-4.

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A143109 Let H(2,d) be the space of polynomials p(x,y) of two variables with nonnegative coefficients such that p(x,y)=1 whenever x + y = 1. a(n) is the number of different polynomials in H(2,d) with exactly n distinct monomials and of maximum degree minus two, i.e., of degree 2n-5.

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A387029 Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n+1 states and of degree 2n-3.

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A386841 Triangle read by rows: T(n,k) is the number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on the n-dimensional probability simplex and of degree 2n-k (n>=1, 1<=k<=n).

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A386840 Number of fundamental one-dimensional discrete statistical models with rational maximum likelihood estimator supported on n states and of degree 2n-6.

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